Mathematical Modeling of Shock-Wave Processes in Condensed Matter

<p>Chapter 1 Models of continuum mechanics and their deficiencies <br>1.1 Description of macroscopic systems; macroscopic variables <br>1.2 Macroscopic transport equations <br>1.3 The problem of closure of the transport equations <br>1.4 Validity of continuum mechanics <br>1.5 Scale averaging effect on transport processes <br>1.6 Medium models and transient processes <br>1.7 The problem of a uniform description of the media motions <br>1.8 Deficiencies of the continuum mechanics concept <br>1.9 Short review of approaches to extension of continuum mechanics </p><p>Chapter 2 Specific Features of Processes Far from Equilibrium <br>2.1 Experimental difficulties in studying non-equilibrium processes <br>2.2 Anomalous medium response to strong impact <br>2.3 The internal structure effects <br>2.4 Fluctuations, oscillations, instabilities <br>2.5 Multi-scale energy exchange between various degrees of freedom <br>2.6 Multi-stage relaxation processes <br>2.7 Finite speed of disturbances propagation and the delay effects <br>2.8 Influence of the loading duration and inertial effects <br>2.9 Dynamic self-organization of new internal structure in open systems <br>2.10 Predictive ability of modeling non-equilibrium processes </p><p>Chapter 3 Macroscopic Description in Terms of Non-equilibrium Statistical Mechanics <br>3.1 Fundamentals of statistical mechanics <br>3.2 Description of macroscopic systems from the first principles <br>3.3 Main problem of non-equilibrium statistical mechanics <br>3.4 Rigorous statistical approaches to non-equilibrium processes <br>3.5 Non-equilibrium statistical operator by Zubarev <br>3.6 Bogolyubov’s hypothesis of attenuation of spatiotemporal correlations <br>3.7 The nonlocal thermodynamic relationships with memory between the conjugate macroscopic fluxes and gradients <br>3.8 Two type of the nonlocal effects <br>3.9 The disadvantages and new opportunities to close transport equations for high-rate processes </p><p>Chapter 4 Thermodynamic Concepts Out of Equilibrium <br>4.1 Basic concepts and principles of thermodynamics <br>4.2 Linear thermodynamics of irreversible processes <br>4.3 Revision of the generally accepted thermodynamic concepts out of equilibrium <br>4.4 Local entropy production near and far from equilibrium <br>4.5 Total entropy generation and the second law of thermodynamics <br>4.6 Maximum entropy principle by Jaynes <br>4.7 Thermodynamic temporal evolution out of equilibrium <br>4.8 Influence of the constraints imposed on the system <br>4.9 Self-organization of new structures in thermodynamics </p><p>Chapter 5 New Approach to Modeling Non-equilibrium Processes <br>5.1 Generalized constitutive relationships based on non-equilibrium statistical mechanics <br>5.2 Modeling spatiotemporal correlation functions <br>5.3 Temporal stages of the correlation attenuation <br>5.4 Deficiencies of the generally accepted models for the medium with complicated properties <br>5.5 Requirements to new approach to modeling shock-induced processes <br>5.6 Foundations of new approach to modeling transport processes far from equilibrium <br>5.7 New approach to modeling transport processes far from equilibrium <br>5.8 Distinctive features of new approach from semi-empirical models <br>5.9 Interrelationships between spatiotemporal correlations and dynamic structure of the system <br>5.10 Modeling correlation functions in boundary-value problems <br>5.11 Boundary conditions for nonlocal equations <br>5.12 The mathematical basis for the self-consistent problem formulation <br>5.13 Discrete size spectrum of the dynamic structure of a bounded system </p><p>Chapter 6 Description of the Structure Evolution Using Methods of Control Theory of Adaptive Systems <br>6.1 Methods of control theory in physics. Cybernetical physics <br>6.2 Speed gradient principle by Fradkov for non-stationary complex systems <br>6.3 Description of the system temporal evolution at macroscale level <br>6.4 Temporal evolution of statistical distributions at microscale <br>6.5 The need to describe temporal evolution out of equilibrium at mesoscale <br>6.6 Principle of maximum entropy by Jaynes and the goal function of the structure evolution <br>6.7 Integral entropy production and reduction of irreversible losses due to self-organization <br>6.8 Internal control at mesoscale based on Speed gradient principle <br>6.9 Paths of the system evolution and prediction of the limit states <br>6.10 Influence of feedbacks on the paths of the system evolution </p><p>Chapter 7 The Shock-Induced Planar Wave Propagation in Condensed Matter <br>7.1 Thermodynamic properties of solids <br>7.2 Wave processes in crystal lattice <br>7.3 Elastic properties of solids <br>7.4 Plastic deformation. Deficiencies of continuum mechanics <br>7.5 Shock wave as a non-equilibrium transient process <br>7.6 The integral model for the stress tensor without separation into elastic and plastic parts <br>7.7 Integral formulation of the problem of the shock-induced wave propagation in condensed matter <br>7.8 Self-similar quasi-stationary solution to the problem <br>7.9 The relaxation model of shock-induced waveforms during propagation <br>7.10 Comparison of the model waveforms with experimental data </p><p>Chapter 8 Evolution of Waveforms during Propagation in Solids <br>8.1 Entropy production in finite-time waveforms <br>8.2 Speed gradient principle for the waveforms evolution <br>8.3 The waveform evolution during quasi-stationary wave propagation <br>8.4 Paths of the waveform evolution on a surface of the entropy production over a phase plane <br>8.5 Coincidence with experimental results </p><p>Chapter 9 Abnormal Loss or Growth of the Wave Amplitude <br>9.1 Dependence of the waveform amplitude on the impact velocity <br>9.2 The wave amplitude loss due to various relaxation effects <br>9.3 Interference of shock wave at mesoscale <br>9.4 Wave packet spreading <br>9.5 Mass velocity dispersion and turbulent effects <br>9.6 Multi-scale momentum and energy exchange in wave processes <br>9.7 Self-organization and the structure instability </p><p>Chapter 10 The Stress-Strain Relationships for the Continuous Stationary Loading <br>10.1 Difference between shock and continuous loading at the same amplitude <br>10.2 Influence of the relaxation and delay effects on the medium response to short and long loading <br>10.3 Entropy production surfaces for various duration loading and possible evolutionary paths <br>10.4 Meta-stable states and the system structural instability <br>10.5 Probable change of the evolution paths and their direction <br>10.6 Dependence of final states on the initial and loading conditions <br>10.7 Influence of the feedbacks between the structure evolution and the material response <br>10.8 Control of the evolution paths to obtain the desired structure of the material <br></p>